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Evaluation of Some Non-Elementary Integrals Involving Sine, Cosine, Exponential and Logarithmic Integrals: Part I

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Evaluation of Some Non-Elementary Integrals Involving Sine, Cosine, Exponential and Logarithmic Integrals: Part I View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Ural Mathematical Journal (UMJ) URAL MATHEMATICAL JOURNAL, 2018, Vol. 4, No. 1, pp. 24–42 DOI: 10.15826/umj.2018.1.003 EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I Victor Nijimbere School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada [email protected] β α Abstract: The non-elementary integrals Siβ,α = R [sin(λx )/(λx )]dx, β ≥ 1, α ≤ β +1 and Ciβ,α = R [cos (λxβ )/(λxα)]dx, β ≥ 1, α ≤ 2β + 1, where {β, α} ∈ R, are evaluated in terms of the hypergeometric functions 1F2 and 2F3, and their asymptotic expressions for |x|≫ 1 are also derived. The integrals of the form n β α n β α R [sin (λx )/(λx )]dx and R [cos (λx )/(λx )]dx, where n is a positive integer, are expressed in terms Siβ,α and Ciβ,α, and then evaluated. Siβ,α and Ciβ,α are also evaluated in terms of the hypergeometric function 2F2. And so, the hypergeometric functions, 1F2 and 2F3, are expressed in terms of 2F2. The exponential integral λxβ α x Eiβ,α = R (e /x )dx where β ≥ 1 and α ≤ β + 1 and the logarithmic integral Li = Rµ dt/ ln t, µ > 1, are also expressed in terms of 2F2, and their asymptotic expressions are investigated. For instance, it is found that for x ≫ 2, Li ∼ x/ln x + ln(ln x/ln 2) − 2 − ln 2 2F2(1, 1; 2, 2; ln 2), where the term ln (ln x/ln2) − 2 − ln 2 2F2(1, 1; 2, 2; ln 2) is added to the known expression in mathematical literature Li ∼ x/ln x. The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given. Key words: Non-elementary integrals, Sine integral, Cosine integral, Exponential integral, Logarithmic integral, Hyperbolic sine integral, Hyperbolic cosine integral, Hypergeometric functions, Asymptotic evaluation, Fundamental theorem of calculus. 1. Introduction Definition 1. An elementary function is a function of one variable constructed using that variable and constants, and by performing a finite number of repeated algebraic operations involving exponentials and logarithms. An indefinite integral which can be expressed in terms of elementary functions is an elementary integral. And if, on the other hand, it cannot be evaluated in terms of elementary functions, then it is non-elementary [6, 10]. Liouville 1938’s Theorem gives conditions to determine whether a given integral is elemen- tary or non-elementary [6, 10]. For instance, it was shown in [6, 10], using Liouville 1938’s The- orem, that the integral Si1,1 = (sin x/x)dx is non-elementary. With similar arguments as in [6, 10], One can show that Ci = (cos x/x)dx is also non-elementary. Using the Euler formulas 1,1 R e±ix = cos x i sin x, and noticing that if the integral of a function g(x) is elementary, then both ± R its real and imaginary parts are elementary [6], one can, for instance, prove that the integrals Si = [sin (λxβ)/(λxα)]dx, β 1, α 1, and Ci = [cos (λxβ)/(λxα)]dx, where β 1 and β,α ≥ ≥ β,α ≥ α 1, are non-elementary by using the fact that their real and imaginary parts are non-elementary. ≥ R R The integrals [sinn (λxβ)/(λxα)]dx and [cosn (λxβ)/(λxα)]dx, where n is a positive integer, are also non-elementary since they can be expressed in terms of Si and Ci . R R β,α β,α To my knowledge, no one has evaluated these integrals before. To this end, in this paper, for- mulas for these non-elementary integrals are expressed in terms of the hypergeometric functions 1F2 and F whose properties, for example, the asymptotic expansions for large argument ( λx 1), 2 3 | |≫ Some non-elementary integrals of sine, cosine and exponential integrals type 25 are known [9]. We do so by expanding the integrand in terms of its Taylor series and by integrat- ing the series term by term as in [7]. And therefore, their corresponding definite integrals can be evaluated using the Fundamental Theorem of Calculus (FTC). For example, the sine integral B sin (λxβ) Si = dx, β 1, α β + 1, β,α (λxα) ≥ ≤ AZ is evaluated for any A and B using the FTC. λxβ α On the other hand, the integrals Eiβ,α = (e /x )dx and dx/ ln x, are expressed in terms of the hypergeometric function F . This is quite important since one may re-investigate the 2 2 R R asymptotic behavior of the exponential (Ei) and logarithmic (Li) integrals [3] using the asymptotic expressions of the hypergeometric function 2F2 which are known [9]. Some other non-elementary integrals which can be written in terms of Eiβ,α or dx/ ln x are βx also evaluated. For instance, as a result of substitution, the integral eλe dx is written in terms β R of Ei = (eλx /x)dx and then evaluated in terms of F , and using integration by parts, the β,1 2 2 R integral ln(ln x)dx is written in terms of dx/ ln x and then evaluated in terms of F as well. R 2 2 Using the Euler identity e±ix = cos(x) i sin(x) or the hyperbolic identity e±x = cosh(x) R R± ± sinh(x), Siβ,α and Ciβ,α are evaluated in terms Eiβ,α. And hence, the hypergeometric functions 1F2 and 2F3 are expressed in terms of the hypergeometric 2F2. This type of integrals find applications in many fields in Science and Engineering. For instance, in wireless telecommunications, the random attenuation capacity of a channel, known as fading capacity, is calculated as [11] ∞ 1 1 C = E[log (1 + P H 2)] = log (1 + Pξ)e−ξdξ = e1/P E ( ) E , fading 2 | | 2 ln 2 1,1 ∞ − 1,1 P Z0 where the fading coefficient H is a complex Gaussian random variable, and E X 2 P is the | | ≤ maximum average transmitted power of a complex-valued channel input X. In number theory, the prime number theorem states that [3] x dx π(x) Li(x)= , µ> 1, ∼ ln x Zµ where π(x) denotes the number of primes small than or equal to x. Moreover, there are applications of sine and cosine integrals in electromagnetic theory, see for example Lebedev [5]. Therefore, it is quite important to adequately evaluate these integrals. For that reason, the main goal of this paper is to evaluate non-elementary integrals of sine, cosine, exponential and logarithmic integrals type in terms of elementary and special functions with well known properties so that the fundamental theorem of calculus can be used so that we can avoid to use numerical integration. β α Part I is indeed devoted to the cases Siβ,α = [sin (λx )/(λx )]dx, β 1, α β + 1, β α λxβ α≥ ≤ Ciβ,α = [cos (λx )/(λx )]dx, β 1, α 2β +1 and Eiβ,α = (e /x )dx where β 1, ≥ ≤ R β α ≥ α β + 1, where β, α R. The other cases Siβ,α = [sin (λx )/(λx )]dx, β 1, α>β + 1, ≤ R { } ∈ R β ≥ Ci = [cos (λxβ)/(λxα)]dx, β 1, α> 2β +1 and Ei = (eλx /xα)dx where β 1, β,α ≥ R β,α ≥ α > β + 1, where β, α R, which may involve series whose properties are not necessary known R { } ∈ R will be considered in Part 2 [8]. Before we proceed to the objectives of this paper (see sections 2, 3, 4 and 5), we first define the generalized hypogeometric function as it is an important mathematical that we are going to use throughout the paper. 26 Victor Nijimbere Definition 2. The generalized hypergeometric function, denoted as pFq, is a special function given by the series [1, 9] ∞ (a ) (a ) (a ) xn F (a , a , , a ; b , b , , b ; x)= 1 n 2 n · · · p n , p q 1 2 · · · p 1 2 · · · q (b ) (b ) (b ) n! n=0 1 n 2 n q n X · · · where a , a , , a and ; b , b , , b are arbitrary constants, (ϑ) = Γ(ϑ + n)/Γ(ϑ) (Pochham- 1 2 · · · p 1 2 · · · q n mer’s notation [1]) for any complex ϑ, with (ϑ)0 = 1, and Γ is the standard gamma function [1, 9]. 2. Evaluation of the sine integral and related integrals 1 3 3 λ2x2 Proposition 1. The function G(x)= x F ; , ; , where F is a hypergeometric 1 2 2 2 2 − 4 1 2 sin (λx) function [1] and λ is an arbitrarily constant, is the antiderivative of the function g(x)= . λx Thus, sin (λx) 1 3 3 λ2x2 dx = x F ; , ; + C. λx 1 2 2 2 2 − 4 Z P r o o f. To prove Proposition 1, we expand g(x) as Taylor series and integrate the series term by term. We also use the gamma duplication formula [1] 1 1 1 Γ(2α) = (2π)− 2 22α− 2 Γ(α)Γ α + , α C, 2 ∈ the Pochhammer’s notation for the gamma function [1], Γ(α + n) (α) = α(α + 1) (α + n 1) = , α C, n · · · − Γ(α) ∈ and the property of the gamma function Γ(α+1) = αΓ(α) (eg., Γ (n + 3/2) = (n + 1/2) Γ (n + 1/2) for any real n).
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